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1 UNCLASSIFIED AD ARMED SERVICES TECHNICAL INAON AGENCY ARLINON HALL SATAION ARLINGIN 12, VIRGINIA UNCLASSIFIED
2 NOTICE: When government or other drawings, specifications or other data are used for any purpose other than in connection with a definitely related government procurement operation, the U. S. Government thereby incurs no responsibility, nor any obligation whatsoever; and the fact that the Government may have formulated, furnished, or in any way supplied the said drawings, specifications, or other data is not to be regarded by implication or otherwise as in any manner licensing the holder or any other person or corporation, or conveying any rights or permission to manufacture, use or sell any patented invention that may in any way be related thereto.
3 R63SD4 MAG NETO HYDRODYNAMIC BOUNDARY LAYERS * A. SHERMAN SPACE SCIENCES LAJ30RATORY cftq GENERAL* ELECTRIC MISSILE AND SPACE DIVISION *THIS WORK WAS SPONSORED BY THE AIR FORCE OFFICE OF SCIENTIFIC RESEARCH, OFFICE OF AEROSPACE RESEARCH, UNDER CONTRACT AF49(638)914
4 SPACE SCIENCES LABORATORY AEROPHYSICS SECTION MAGNETOHYDRODYNAMIC BOUNDARY LAYERS* by A. Sherman This work was sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, under Contract AF 49(638)914. R63SD4 January, 1963 MISSILE AND SPACE DIVISION ELECTRIC
5 CONTENTS PAGE 12.1 Introduction The Rayleigh Problem Formulation of Boundary Layer Equations Incompressible Boundary Layer Flows Compressible Boundary Layer Flows Magnetic Boundary Layers 29
6 FOREWORD This report has been written as Chapter 1Z of a forthcoming book, Engineering Magnetohydrodynamics. References to other chapters refer to other chapters in that book.
7 12.1 INTRODUCTION The subject matter to be covered in the present chapter will consist of Magnetohydrodynamic flows of a boundary layer character. In conventional fluid dynamics, boundary layer theory has been developed to a high degree of sophistication. No comparable extensive treatment will be attempted here. The problems which will be discussed will be chosen to illustrate those new features introduced by the interaction between currents flowing in the conducting fluid or plasma and the electromagnetic field. The principle interest in the present chapter will be in flows for which the magnetic Reynolds number is vanishingly small. For completeness, however, the boundary layer character of flows with Rm  will be treated in the last section. The first problem considered will be the Rayleigh problem in which a magnetic field is applied normal to the surface of an impulsively moved half plane. Its principle value will be in that a solution can be obtained in closed form so that the nature of Magnetohydrodynamic boundary layer flows can be inferred. Following this introductory treatment the boundary layer equations are obtained from the full equations by means of the well known boundary layer approximation. Based on these equations, incompressible boundary layer flows are examined first. Two methods of solution of the nonlinear partial differential equations are developed. For flows which satisfy certain specified conditions, similarity exists and the equations can be reduced to an ordinary differential equation which must then be integrated numerically. For more realistic boundary conditions similarity cannot exist so that in this instance series solutions are necessary. For high speed flows (hypersonic) compressibility effects must be considered, so compressible boundary layers are considered next. In this case the boundary conditions are such that similarity conditions can be applied and the solution to the problem reduces to that of numerical integration of two coupled ordinary differential equations. To conclude this chapter the electric current and induced magnetic field boundary layers which can form on surfaces when the magnetic Reynolds number is very high are considered. Although the Magnetic Reynolds is extremely small in almost all cases of interest in continuum Magnetohydrodynamics the above subject is of interest since it involves application of old boundary layer concepts to a new physical problem.
8 12.2 THE RAYLEIGH PROBLEM The problem which will be studied in the present section is shown in figure BO ft??t Figure Rayleigh Problem Configuration A field B is applied perpendicular to the plate and is assumed to be uniform in space and constant in time. to move in x direction impulsively at t = 0. The doubly infinite plate (a nonconductor) is assumed The problem is to determine the induced flow and magnetic field due to the impulsive motion in general. A number of authors have investigated the general problem( 1 ) ' (2) and have calculated the coupled flow and induced magnetic fields by means of various approximations. which Rm << Re. Of principle interest in the present chapter will be flows in In this case the problem reduces to one in which the flow disturbance is confined to a thin boundary layer region while the induced field varies over a much larger region. The equations appropriate to the present problem are essentially the same as those for the transient couette flow, wall is infinitely. far away: except that now the second, stationary, au B abx 2 P T 0 y + 7 (12.1) i~2 (3bx a t au Be ab2 b (12.2) 0 ay a ay 2 The boundary and initial conditions are t = 0: u(y, 0) 0 bx (y, 0) 0 t > 0: u (0, t) =u 0 b x (0, t)= 02
9 where, of course, u and b must remain finite as y  ao. The requirement that b x = 0 on the moving nonconductive wall is arrived at in much the same way as was done for Couette flow, see Chapter 10 section The above equations can be most readily treated with the aid of the Laplace transform defined by L if (y, t)1 =? (y, s) f 00e  s t f (y, t) dt 0 where s is some constant parameter. Taking the Laplace transform of (12. 1) and (12.2) gives s B A d2a B + v d (12.3) p I dy dy 2 A d2 dau 1 x (12.4) x dy dy 2 where v = i7/p = kinematic viscosity and the boundary conditions are now A u (o, S) = u2 (0, S) 0 The solution of equations (12.3) and (12.4) is readily found to be m 2 n2 A (s) (S) ya emy + BemB n y (12.5) 0 n B o = Ae  m y + Be  n y (12.6) where A and B are arbitrary constants. been omitted, and +m and ±n are solutions of which is satisfied if Terms which diverge at infinity have r 2 2 Bo 2 (  s) (vr S) r 2 =0 (12.7) ob 2 ctb 2 m= +as a + + S = a B02 a ab OB0 B 2 +f s  )1 s 3
10 where (a 1 _"VT 2 V9_ The constants A and B must be selected to satisfy the boundary conditions imposed. The solution then assumes the following form: 2 2 A 1 ( " in ) (n u m ( 5s) n m [lm. s ] m r nny (12.8) A b X U0 e 3m  enyf (12.9) B 0 s (mn) L + 5] Inversion of these equations is quite difficult as they stand. Since our principle interest will be in flows in which Rm 0, and this implies P m  0 or 1 a solution of boundary layer character can be In V!Reu anticipated and the problem accordingly simplified. Keeping up finite, and letting v  0 while holding y/iv  fixed leads to the following results: 1/2 02+B2]/ v] n s [L + PA' and A 1 e S + y _ 1 e v (12.10) u00 s A bx= 0 Accordingly, when b x = 0 at the moving wall it is zero throughout the velocity boundary layer region. Such a result could have been anticipated. If on, the other hand, other boundary conditions had been selected for b x other results would have been obtained. times.b x (0, If the problem geometry were such that for long t)  constant rather than zero then the boundary condition would be that b x at the boundary be some function of time depending on the total current flow. Such a boundary condition could not be conveniently handled. Finally, if the moving wall is assumed to be a perfect conductor the electric field in the fluid adjacent to the wall must be zero. Then from equations (10. 1) and (10. 5a) 4
11 we have or  a_ Ez + ubo al z 0J ab x Ely (o,t)=  0uB o (12.11) which is the required boundary condition for this case. As has been shown by Ludford 11) the solution when equation (12. 11) is used as the boundary condition leads to a b x which is constant throughout the boundary layer region. Returning now to the problem originally posed, the inverse of equation (12. 10) is readily found to be _1[e Y erfc   p t / 2 + e erfc Y + (12.12) This result is identical to that of Rossow 12 ) although the induced magnetic field was not considered in that reference. as b x = 0 throughout the boundary layer region. The reason for the agreement is obvious However, it should be noted that b x will become finite above the boundary layer and approach some constant value at . The value of formulating the complete problem before making the boundary layer approximation will be seen shortly when the shear stress at the wall is calculated. First, however, consider the solution for u found above. If B is allowed to go to zero equation (12.12) reduces to the classical Rayleigh problem: '2' = 1  erf(  \ (12.13) In this case the velocity profile depends on only one variable 2 which is essentially the similarity variable to be discussed later. In the presence of magnetohydrodynamic effects such similitude no longer exists and there is no one variable on which u_. depends. In order to illustrate the flow pattern, calcu U00 lations based on equation (12.12) have been carried out and are presented in Figure
12 2.5 w I J  * DIRECTION OF PLATE CL MOTION. > 1.5 DIRECTION OF APPLIED o0 MAGNETIC FIELD P z 0.0 O 0.2 OA U VELOCITY u Figure Velocity Profiles in the Rayleigh Problem It can be seen that application of the magnetic field increases the time required for the flow velocity above the plate to reach any specific value. If desired, the above solution can be interpreted as a boundary layer flow with t  x/ua. In this case we observe that as the field strength is increased the velocity profile becomes fuller. Hartmann flow earlier. This result is analagous to that found for the Also, the above results suggest that a natural variable to formulate the boundary layer problem with should be T) = y just as in the in the case of conventional incompressible boundary layers. Finally, the force on the moving plate will be considered. is given simply as The shear stress (l") = T( a W)y 'Yy=O (12.14) Now from equation (12.8) the transform of T. may be derived cib 2 A Vu m T= (12.15)
13 The inverse of this expression gives the following result TW (t) " Ie0 1  t +/Y erfv/it} (12.16) where CB 2 P (1+rp m )7 which agrees with Rossows ( 2 ) result when Pm = 0. The above equation is valid for arbitrary values of Pm and is thus a much more general result. In order to illustrate the influence of Pm on the solution, equation (12.16) is plotted in figure A number of valuable observations can be made based on these results. First, it is clearly seen that as a increases the wall shear becomes larger for any given time. This increased shear arises as a result of the enhanced Lorentz force on the conducting fluid near the moving wall. At very long times all of the curves of Figure 122 reach asymptotic values given by lim t  c (TW')= Now in the nonmagnetohydrodynamic Rayliegh problem equation (12.13) shows that u co as t   so that TW  0, and it is of interest to inquire as to why in the present problem TW' # 0 in the same limit. If we return to the original equations of the problem, equations (12. 1) and (12. 2), and assume  = 0 the at following solution is obtained for u and b X. B 2 u = e U00 (TB 0 y bx Va1 ucl [A  i The steady state long time solution then has an exponential velocity profile rather than the constant one when say a = 0. This shape of the profile arises because of the existence of the currents in the fluid and the attendent Lorentz force. The form of the current distribution is also exponential decreasing from some finite value at the wall to zero at infinity. 7
14 '2 t Figure Wall Shear in Rayliegh Problem
15 From the point of view of the physical problem the wall shear is balanced by the reaction force on the source of the magnetic field. When there is no magnetic field there is nothing to balance this shear and so it must be zero in the steady state FORMULATION OF BOUNDARY LAYER EQUATIONS The boundary layer approximation in fluid mechanics has been discussed by many authors from the time of Prandtl; there is, therefore, no justification in repeating these arguments. What will be discussed in the present section will be those new terms which appear in the equations due to Magnetohydrodynamic effects, and their simplification via the boundary layer approximation. Attention for the present will be restricted to flows with Rm  0 so that the magnetic field can be taken as the applied field (i.e. section 10.8 of chapter 10). The question of the boundary layer approximation when Rm   will be discussed later. The general problem which will be considered is shown in Figure The magnetic field vector is assumed to lie in the xy plane but may have x and y components. The Lorentz force for such a two dimensional problem is given by F = i_ LB 0 = [E + q xb] xb (12.17) B Figure Magnetohydrodynamic Boundary Layer Problem 9
16 where the current is derived from the Ohm's law since induced magnetic fields have been assumed negligible. Now since only steady flow problems will be considered it will be valid to take E = 0 or if desired some constant. It may be recalled that in the unsteady problem (i.e. transient Couette flow) it is not in general valid to assume E = 0. written as For the present case, equation (12. 17) can be F x = ao(uby  vb x B) (12.18) Fy = a (ub x B y  vb x2) (12.19) For purposes of order of magnitude arguments one can assume that B = 0 (Bx). Then, since v = (6) where 6 is the boundary layer thickness the first of the above expressions can be written as Fx n:  By 2 u (12.20) where By can be a function of x and y in general. In a boundary layer calculation it will be exceedingly inconvenient to keep By a function of y. Accordingly, it is of interest to study By (x, y) somewhat more carefully. For a two dimensional field one of Maxwell's equations (VB = 0) requires 8B 8B  + Y =0 i8x axa EBy  1 ) Now with B x = C9(By) and (l) (1= while ( 6= we have. y ax ay or aby ay 0(1) ABy = 0 (6) A y Thus, the change in By across the boundary layer will be the order of 6 and can be neglected. Making use of equation (12.20), assuming p = p(x) and making the usual boundary layer approximations the x component of the momentum equation becomes: becomes: p (U au + v au+ = B 2 u + u) (12.21) Before proceeding further the assumption that p = p (x) must be examined further. Again assuming B = (Bx) equation (12.19) reduces to 10
17 F  acub B (12.22) y y x Now the y component of the momentum equation can be written in the following form Pu + pv Lv + = QuB B + (12.23) ax ay ay x y Re (y a where YBR2 L PR U00 R PR U00 L R =~ e IR and the velocities have been made dimensionless by some reference velocity (uoo), 2 the distances x and y by some reference length (L), the pressure by pou 0 O, the density p and viscosity 1 by reference values, and the magnetic field by some reference field (BR). For magnetohydrodynamic boundary layer problems of interest Q = 0 (1). Then an order of magnitude analysis of equation (12. 23) with u = )(1), v = (6), L = 0(1), ay 0 (1), and e = () gives or ay Ap = ((6) ~ 0 Thus, the pressure change across the boundary layer is the order of 6 (a small quantity) and can be neglected. It should be noted, however, that in the absence of magnetic forces A p = 0)(62) so that it is less valid to assume p = p(x) in the present case than it had been in the absence of a magnetic field. Next, the Joule heating term which will enter into the energy equation must be considered..2 H = _= T(xB) (qxb) (12.24) or H= a 2 +v2bx 2uvB xbyj as before, the last two terms can be neglected since v/u = () (6). Thus, H au 2 B 211
18 and the energy equation can be written as follows P h h udp 8x  + v ~  dx= 8 h + _. kj (Y P R ay) ay (e p + O 2 B y 2 + (au' ay/(2.5 2 (12.25) where R = Cp7 k Finally, the mass continuity equation must be added to complete the system of equations: T (Pu) + ay (Pv) = 0 (12.26) Before proceeding to the solution of specific problems it will be of interest to study the general question of similar solutions for incompressible, constant property, boundary layer flows. free stream flow p+ +oby dx dx Consider equation (12.21) evaluated in the inviscid 2 u00 (12.27) Combining this with equation (12.21) leads to the following relation. au au du 0 0 o B 2 a 2 u u  + v  uo + ab 2(uu0) = avu (12.28) ax ay dx P 8y 2 Introducing u = and v =  a o satisfies continuity and equation (12.28) Inroucng El y a x becomes2 becomes B U00 d uo + v yy (12.29) 0y yx x yy + p y  U00) dx yyy The problem now is to determine whether or not the above can be reduced to an ordinary differential equation by a proper choice of variables. 4 (x,y) = UV i x f (n) Let us assume where t7(x,y) = Y and where it should be noted that uoo and By are known functions of x as yet unspecified. Substituting the above equation (12. 29) becomes 12
19 2! (f')  + d ff,, P )= +F X ' p (11 fl)  8du +  (12.30)  B 2 f 1 Accordingly, similarity can be achieved if and duco u dx x so that u 0 o and B must be of the following form to permit a similar solution U= C1 xm m1 B = C 2 x 2 In fluid mechanics the xm variation of uo corresponds to the wedge flow solutions. In magnetohydrodynamic problems such a simple interpretation is not possible since the applied magnetic field interacts with the inviscid free stream flow. We will return to this point in the next section. A final point in regard to the above treatment should be made. In looking for similar solutions only the incompressible boundary layer was studied. For a compressible flow the situation is much more complex. Now, for a similar solution to be found for the compressible boundary layer problem it is safe to say that a minimum condition would be that ueo and By satisfy the conditions already established for the incompressible case. In addition, many other assumptions and requirements will be necessary to obtain similarity in the compressible case. Some of the new phenomena associated with compressible magnetohydrodynamic boundary layer flows will be treated in section The following section will be restricted to incompressible boundary layers INCOMPRESSIBLE BOUNDARY LAYERS The distinguishing feature of an incompressible magnetohydrodynamic boundary layer is that the inviscid flow external to the boundary layer is also a conductor. Then, since it is not possible to restrict the magnetic field to the boundary layer region alone, one has no choice but to incorporate the results of the inviscid magnetohydrodynamic analysis into the boundary layer 13
20 investigation. On the other hand, this may not be necessary in compressible flows in which the conductivity can vary, and for certain types of flows may be assumed zero external to the boundary layer. Application of the similarity solution to incompressible boundary layer although feasible, in principle, will not be carried out due to the difficulty of interpreting the resulting inviscid flow. Instead, two simple incompressible boundary layer problems will be analyzed by the series expansion technique Boundary Layer with Uniform Free Stream The first problem that will be considered will take as a model for the inviscid flow the flow through a parallel walled two dimensional channel and will consider the flat plate situated somewhere in the channel some distance from either wall. In this case, the free stream velocity will be a constant and the pressure gradient will be given by dp/dx = a u o B 2. The applied magnetic field will be assumed constant so that the pressure will vary linearly along the channel. The equation to be solved is equation (12.29) which simplifies to the following. ub 2 yyx  xyy + ( u) yyy (12.31) where the boundary conditions are 0 = y =0 at y = 0 4, y uoo at y = , x = 0 Since the pressure gradient in the present problem is constant and favorable it will be adequate to use a classical Blasius series expansion to effect a solution. Assume r crby2, p_ \2 1 A(x,y) = I fo + p x fl + x) f ] (12.32) where it is assumed that (P 9 x) is a small quantity and the f's are functions of 71 = y. Introduction of the above assumption for 4 into equation (12.31) leads to the following infinite set of ordinary differential equations. fo" +f fo " = 0, (12.33) with the following boundary conditions: fo () = f 0 ' (0) = 0 fo'(l) 114
21 f, f fo ff  fff fo' fo f f0 0 o 1 (12.34) with with f 1 (o)= fl' (o) 0 fl' (1 )= 0 "I+1,, f2" 2fI + 5 fo itf= 3 ffl"+f f 2 ' + f o  2 fo' f 2 ' 2 f 2 = (f1') 2  f + f 2 (o) =f2' (0) =0 f2' ( =0 (12.35) etc. The first of these is the well known Blasius equation for which the solution has been tabulated for the boundary conditions shown. All subsequent equations are linear, but since they depend on the preceeding solutions numerical integration is necessary. The first two equations beyond the Blasius have been solved by Rossow (2 ). The resulting velocity profiles are shown in figure ST ORDER IN Woo X NO ORR IN r X 4 P Figure Incompressible Boundary Layer for Uniform Applied Magnetic Field and Free Stream Velocity Again these results are in agreement with the Hartmann flow treated in Chapter 9 and the Rayliegh problem solution discussed earlier in the present chapter. In other words, the presence of the applied magnetic field tends to make the velocity profile fuller. If, on the other hand, the free stream velocity had been decreasing with x the pressure gradient would have become unfavorable, the boundary layer profile would have become less full, and the possibility of flow separation would exist. Such a case will be considered next. 15
22 Boundary Layer Subject to Adverse Pressure Gradient Consider, as an appropriate inviscid flow the flow through a parallel walled channel when the applied magnetic field is created by a currentcarrying wire alligned perpendicular to the flow direction and imbedded in the lower wall. As shown in Figure 126 the boundary layer development along the lower wall will be considered. UNIFORM ENTERINGA FLOW.y Y S:] MAGNETIC LINES OF FORCE BOUNDARY LAYER WIRI Figure Boundary Layer Development with Magnetic Field Generated by CurrentCarrying Wire in Lower Wall One must, of course, assume that the boundary layer ahead of x = 0 can be removed, perhaps by a bleed port shown schematically as the dotted region. The equation and boundary conditions for 0 have already been presented, equation (12.31), and will also be used in the present problem. One should be aware, however, that requiring Oy = uo at y =  shear (see Figure 1033) at the lower wall is being neglected. means that the inviscid flow In general, this shear can be neglected when the inviscid free stream flow has been linearized. As noted earlier, the existence of a nonuniform free stream velocity, and the possibility of separation, requires the use of a series solution which is more sophisticated. A procedure frequently used for conventional boundary layers and ideally suited to the present problem is due to Gortler. Its extension to magnetohydrodynamics and its application to the present problem will be carried out now. If the following dimensionless variables are defined y   YO YO ui ", ]ReQ_ yxu y B0 BI, U0 Y o u i y Bo 16
23 and Re = UiYoV Q = B 0 2 YO where u i = u (o,y), assumed constant, y 0 is some reference length, and B o some reference magnetic field, and the following new independent and dependent variables are defined Then equation (12.31) now becomes F77777 FF 72 F 7 2 B 1 24 F F t 7 Ft FTi + (F 1) 2 (12.36) with the following boundary conditions F = F7 = 0 at 17 = 0 F = 1 atq= o The problem can be solved in general by assuming the following expansions: F (t, 7) = F o ( 1 7) + 4 F 1 (7 1 ) + 42 F 2 (11) duo = /(t) = p 0+ t/p1 + t2 P dx 0 ~ 2 M 0+t91+t 2 iy 2 =~ ~ B2 g (4) =go +4g +4 2g U0o Substituting these into equation (12.36) one obtains, as before, an infinite set of equations. The first of these is the Blasius equation Fo" + F o " = 0 (12.37) F o (o) = F (o) = 0 F o ' ( ) 1 The second and all later equations are linear, and are given by the following recursive system = fx,0 = Uo 17
24 F111 + F ofk''  2kFo Fk + (2k + 1) F" F + Rk = k =I, 2, 3, (12.38) where 2 k1 kj k = ki1 J=I i=1 j1 I kij k1 k1 F' 0 Z j=l F j1 kj j= (k j) F1 F' kj ki + 2; i=l 1 + 2j) F F"  + 2gk1 (I Fo) 2 k12 l ~ gj1 FF1_ j=i kj and Fk = F1 = 0 at n = 0 F' k = 0 at?= i. Application of the above equations to the solution of the specific problem of Figure 126 has been carried out by Sherman (3. Numerical results were obtained for an inviscid flow in which L/y o = 1.6 and Q = 0.1. Curves showing the wall skin friction, boundary layer thickness, and velocity profiles are shown in Figures 127 and \ or, 0.. ID 1.i 1.4 r Figure Wall Skin Friction Versus Distance From Leading Edge for Q = 0.1 and L/y o = 1.6. Nonmagnetic case  ; Magnetic Case, Series Solution ; Magnetic Case, Integral Approximation
25 IC 4 2. ' U48 Figure Boundary Layer Velocity Profiles at Several Positions Downstream of Leading Edge for Q = 0.1 and L/y o = 1.6 In order to interpret these results properly it will be of value to review the physical phenomena occuring in the flow. First, the applied magnetic field acts on the inviscid flow in such a way as to cause the boundary layer free stream velocity to decrease rapidly in the vicinity of the wire. This tends to make the velocity profile less full, and thereby reduce the wall skin friction. In addition, the magnetic field within the boundary layer creates a Lorentz force which tends to retard the flow. This effect then also tends to reduce the skin friction, and both combined may retard the boundary layer sufficiently to cause separation. That such flow separation is indeed a practical possibility is shown in Figures 127 where T = 0 at x = 1.22 L, just beyond the wire. For the present strongly retarded flow the classical KarmanPohlhausen integral approach is seen to be a poor approximation. The details of the velocity profiles are shown in Figure 128 where it must be kept in mind that uao is decreasing as x increases from zero COMPRESSIBLE BOUNDARY LAYERS Although some of the requirements for similar magnetohydrodynamic boundary layers had been identified earlier such solutions were not sought in the incompressible case due to the difficulty in interpreting the results in terms of a practical problem. When the flow is compressible, however, such difficulties may not exist. For example, when considering the hypersonic flow over a flat plate in which the free stream is at a low temperature compared to the high temperatures in the boundary layer, one may assume r = 0 so that the applied 19
26 magnetic field does not disturb the inviscid flow and u~o = constant. Also, there are some practically important problems for which the interaction between the magnetic field and the inviscid flow have been calculated and have shown u~o have the required form for similarity. In the cases just cited similar solutions have practical significance, and in fact are absolutely essential for progress to be made in solving these complex problems. In the present section two compressible boundary layer flows will be treated in some detail, and a third will be discussed. First, consider the hypersonic flow over a semiinfinite flat plate when the temperature within the boundary layer is high enough to ionize the gas. to In order to obtain a similar solution, the free stream velocity is assumed constant (low free stream temperature so that a = 0), assumed to vary inversely as the square root of x. and the applied magnetic field By is In addition, the wall temperature is assumed constant, and the gas is considered to be in thermodynamic equilibrium. The geometry and coordinates of Figure 124 will be used. Neglecting heat flux due to diffusion of species (current does not flow across a temperature gradient), the Hall effect, and induced magnetic fields, and putting dp = 0 the boundary layer equations are given by Equations dx (12.21), (12.25), (12.26). The magnetic field can be expressed as and B = The reduction of these equations to the ordinary differential equations corresponding to similarity is accomplished by the Crocco transformation (4 ). be sketched briefly. The procedure will If the independent variables of the problem are changed from x and y to x and 8u u and T S the transformation formulas can be written as follows: X 8x/y au/ (4)x = ( (8)x = Th ( )x recognizing that we can consider y = y (x, u) the first of these leads to ax au 20
27 or u Ox) yx (Fou ' 'yu and the second yields or  1 YU T Then the transformation equations can be rewritten to yield Using these relations the transformed boundary layer equations become a(pv) a(pu) I E1 (Pu) (12.39) au Yx au T X P+2 Or obo 2 L 71 B02L 7 u (12.40) pv = (Pu) Yx + u (2 Puu T L  ab 2 )W 0+ u+ au o o au + T E ( T h + O2 (12.41) 17 u P R au x 7 where the second term in the last of these has been simplified with the use of equation (12.40). Eliminating pv from equation (12.40) by use of equation (12.39) gives Pu + 7 (Pu)x + 2 B 2 Iu p x dul 20 L O d u T or od2 (Pu7)+ (pau_) +d 2T B2B 2Lo d ( 7u ) = 0 (12.42) T ~ 0 Io du,i r1.2 x du T Next assume that T (x, U) = G(u) and h (x, u) = h (u) 21
28 so that equations (12.41) and (12.42) become Pui _q d 2 G B 0 Lo d ( 2G V V du2  du G 0 or G d 2G Pu + GB L d  d du (12.43) du\n3 and2 G dg dh x du du R x du  B0 2 L o u dh i7x du )+ u BxLx or dg dh d (G + du = TU R du rb 2 ulo t + ] (12.44) GB du These are then two ordinary differential equations for G and h as functions of u. The numerical procedure for their solution is given in detail by Bush (5) where the method of expressing p, 17, cr, and PR as functions of h for high temperature air are also described. Some of the results of calculations for the constant wall temperature case are shown in Figure r/rnm /# 4 I / 4I / UI / 0 I I I I IW, 10 1 / Figure Shear Stress and Heat Flux for Hypersonic Boundary Layer
29 The conditions chosen for the above case were To = 222 K, p = 103 atm, Moo = 25, and a wall temperature of ~2000 K. The subscript NM stands for nonmagnetic, and Q is the interaction parameter proportional to B.2 The above results bear a striking resemblance to the real gas Couette flow results described in Chapter 10. Again, a hysterisis effect exists due to the form of the a versus T curve and the dotted portion of the curves are unstable. The principle new information here is that in allowing the boundary layer to grow in height (in the couette flow 6 is fixed) a reduction in heat flux of 80% is possible where little if any heat flux reduction had been predicted earlier. The shear stress does, however, behave as predicted. Another compressible boundary layer flow of practical interest is one which grows along the electrode surface of a crossed field MHD channel. The principle new feature here is that a current flows normal to the boundary layer surface (see Figure 1210) so that the contribution to the heat flux due to the diffusion of electrons in a temperature gradient cannot be neglected. SELECTRODE SURFACE Figure Electrode Boundary Layer Configuration Since the boundary layer under consideration will be developing within a finite width channel the usual assumption that the boundary layer thickness is not large enough to disturb the inviscid flow must be made. In addition, it is necessary to assume that the electrical resistance of the boundary layer is small compared to the resistance of the inviscid flow so that the overall current flow is determined external to the boundary layer. The momentum equation is given by equation (12.21) except that now the Lorentz force is written as j B since (j) is now some known function of x. Y pu + P + x= (L y) + j B (12.45) 23
30 The energy equation written in terms of temperature (h = cp T) is given by equation (12.25) except that here the Joule heating is written as j2/c to the fact that j = j (x) is a given quantity. equation (12.26). again due The continuity equation is, of course, Since (j) is the same within the boundary layer as in the free stream the momentum equation in the free stream is du S 9 =B (12.46) and combining this with equation (12.45) gives 8u + au =" _ ( u ) + 8X a ay ay pu du, (12.47) dx The energy equation evaluated in the free stream yields dto =.2 POO u0 C (00 u ) which when combined with equation (12.25) yields the following relation 2 ~a(y )u T +5k T)v+ p u vayy 2e + p dx u oo U0. It is interesting to note at this point that the momentum equation for the present case is independent of By. the energy equation. The principle Magnetohydrodynamic effects appear in For convenience in the boundary layer analysis I. and K will be assumed proportional to T, despite the fact that for constant mean free path, kinetic theory leads to a T1/2 dependence. In general (see chapter 5) the electrical conductivity can be determined as a function of pressure and temperature. time, for convenience in both the free stream and boundary layer analysis the pressure dependence will be ignored. so that This Finally, the gas will be assumed perfect p = prt (12.51) Now, the boundary conditions which are needed are Too = To (x) u0o = uo (x) In addition, expressions are needed for p (x) as well as j (x). All four of these relations can be obtained from the solution of the inviscid problem. For the inviscid flow, it will be assumed that Too and Eco are constant, the former also 24
31 leading to a constant aw if the pressure dependence of a is neglected. these assumptions the solution, based on the methods of chapter 11, is Uoc = tx n Within 2 p = oo (RToo Ec) (5n1) 15n (12.52) a2 a 2oRTo a a n where one must have n > 1/5. possible. 2n (5n1) nn x 2n As noted earlier, u., in the above form may make a similarity solution taken up next. and The feasibility of reducing the equations to similar form will be Define the following independent variables x u = f P o dx (12.53) o PO UoC u0 fy dy (12.54) ov C , 0 PO where ( )0 denotes some convenient reference x position. Next, define a stream function in order that mass continuity be satisfied Oy PO Ox pv and then redefine the stream function to be 0 = V V O u 0 V f (t) The momentum equation then becomes fu + t + du (f)2] = 0 (12.55) If a dimensionless temperature is defined as 0  the energy equation becomes 1 a20 + f 22 fv4 L  (y1) M PR t 2 a 2 (f1) 1/2   ] C!Pouoo p
32 2 2 u o P  f0 ) (12.56) ' C T 00 p 00 uc 2 ( "P ' Using the results of the inviscid analysis, equations (12.52), to determine and restricting the value of n so that n <  we find x = (24n) ( ) (12.57) Xo0 1 0 and the energy and momentum equations can be written as follows e P R a7 2 a at a2o 08 1 a + f 8= 2f (l) M'0 2 2 (f,) 1/2 (,)(5n1)1/2 5kTo o, 0 0 n2e 17ORT 0a  (12.58) and f" + ffl 1 + n  0 (f )2] = 0 (12.59) Accordingly, it can be seen that since Mo = Moo (4) that the first two terms on the right hand side of equation (12.58) are functions of 4 and prevent a similar solution. If, however, the wall temperature is constant then 0 # 0 (0, and the 2 first of these terms vanishes. Finally, when Mo =0 the second term can be neglected and a similar solution is feasible if Moo 2 constant and the similarity is then in exact. 0 it can be taken to be a In the present example one can clearly see that when the flow is compressible the specification that uoo a x n is not sufficient to ensure a similar solution, but that additional conditions and assumptions are necessary. Calculations based on the above equations have been carried out by Kerrobrock (6 ) for Helium seeded with Cesium. The free stream temperature and Mach number were taken to be 3000 K and unity respectively. A wall temperature of 1500 K was also chosen. Within the boundary layer being considered heat is generated by two mechanisms: viscous dissipation, and Joule heating. Energy is transported from these two 26
33 sources and from the high temperature free stream to the wall by both conduction and diffusion of electrons against a temperature gradient. For the selection of parameters made by Kerrebrock both viscous dissipation and Joule heating are of comparable importance, and energy transport by electrons is only a few percent of the overall heat transfer. To see more clearly the influence of the above heat sources on the boundary layer, velocity and temperature profiles are shown in Figures n I3 n 0.,1 / / / FLAT PLATEI I u T/T o 0 : U /u C Figure Temperature and Velocity Profiles for Electrode Boundary Layer. M = 1, Increasing values of n correspond to free streams with increasing acceleration. This is seen to correspond to the fact that the velocity profiles tend to become fuller as n increases. The temperature profile marked "flat plate" corresponds to the case for which there is no current flowing at all. The large difference in a Io between the dashed curve and the others is indicative of the increased heat flux due to Joule heating in the low temperature region near the wall. For the numerical example chosen the increase is approximately an order of magnitude. The temperature excess at the highest accelerations can be attributed to heat generated by viscous dissipation which cannot readily transfer to the wall due to the large amount of heat liberated near the wall by Joule heating. The final item of interest in regard to the electrode boundary layer is the electrode potential drop. Due to the fact that the gas in the vicinity of the wall is at low temperature and consequently low electrical conductivity, the electrical field in the vicinity of the wall will be much larger than that in the free stream in order to be consistent with a constant current flow. This will lead to a 27
34 larger potential drop across the boundary layer thickness than across a corresponding thickness of the free stream. Qualitatively, the potential distribution, based on the electric field seen by a stationary observer, across a complete channel will be as shown in Figure h Y STREAM1 BOUNDARY LAYER REGIONS 0 Figure Potential Distribution Across a Channel of Height h. The precise magnitude of the potential drop across the boundary layer can be calculated readily. Consider the potential drop in excess of the potential drop through an equivalent thickness of free stream. Then 6 = f (E E..) dy= f j  a B(uo  u)i dy (12.60) o oo La I where it should be noted that the reduction in a in the boundary layer will tend to increase 60 while the reduction in flow velocity there will tend to reduce it. Depending on the assumed conditions 68 can be positive or negative. of 60 based on the example cited earlier are shown in Figure Calculations Figure O0 M I l n Boundary Layer Potential Excess 28
35 It is interesting to observe that 60 is positive, for this case, so that the potential drop through the boundary layer is indeed greater than in the free stream over the same distance. Finally, mention should be made of the magnetohydrodynamic compressible boundary layer in the region of the stagnation point on a blunt body. The practical example is, of course, the reentering nose cone. For this problem the inviscid flow has been calculated and it has been shown that uoo = ax when B y is a constant, the magnitude of a being reduced as the strength of the magnetic field is increased. These are precisely the minimum requirements for a similar solution, as noted earlier, and with several additional simplifying assumptions similarity solutions can indeed be obtained. (7) 12.6 MAGNETIC BOUNDARY LAYERS Up to this point in the present chapter, and in fact throughout this book, problems of external flow (flow over closed bodies such as airfoils) have not been considered. It will be of interest, however, to consider qualitatively some new boundary layer phenomena that arise in such flows when Rm is large and PRm is small. (8) The particular overall problem that will be investigated will be the so called "alligned flows". These are flows in which the flow velocity and magnetic field vectors far from the body are parallel. Now when Rm = o and the electric field, E, is zero the Ohm's law requires that vxb = 0 so that v and B are not only parallel at infinity but are parallel everywhere. For a body of finite conductivity E = 0 implies j = 0 so that the magnetic field is harmonic within the body. Since it must be a constant on the surface the mean value theorem of potential theory tells us that it must be zero everywhere within the body. Accordingly, the tangential component of B at the surface must jump from a finite value to zero. surface. As was shown in Chapter 2 this corresponds to a current sheet at the vortex sheet there. Since v must also go to zero at the surface there must also be a When Rm is large but no longer infinite these current and vortex sheets are in reality boundary layers. The physical nature of such layers, and the equations governing them will be _Rm the subject of this section. Since, as was seen earlier, P = Re the assumption of large Rm and Rm fe small PRm is tantamount to assuming Rm large and Re much larger. Accordingly, it can be anticipated that the boundary layer in question will really be two layers. One will be an outer layer in which viscosity is negligible, and the other will be a viscous sublayer. The outer layer thickness will be the order of Rm  1 /2, 29
36 1/2 Tefo xenlt h while the inner layer will be the order of Re  The flow external to the outer layer, where v ab, will be irrotational. of the boundary layer region is illustrated in Figure The above physical interpretation POTENTIAL FLOW U INVISCID BOUNDARY LAYER VISCOUS SUBLAYER Figure Sketch Showing Inviscid Magnetohydrodynamic Boundary Layer and Viscous Sublayer Before considering the equations and specific boundary conditions in detail, it will be of value to first discuss the procedure for solution. First, the potential flow must be determined neglecting both boundary layers. The boundary conditions for such a solution are that the surface be both a fluid and magnetic streamline. From such a solution one obtains values of u and B at the wall. These, then, will serve as outer boundary conditions for the inviscid layer. The inner conditions for this layer are evaluated by assuming the viscous layer to be of negligible thickness. Thus, one of the inner conditions on the inviscid layer will be v = 0 while u 0. Another inner boundary condition on the magnetic field will be needed to complete the formulation of the inviscid boundary layer. The viscous layer will then use u (x, o) as obtained from the inviscid layer solution as its outer boundary condition along with v = 0. The inner boundary conditions on the viscous layer are the conventional ones of u = v = 0. So far little has been said about the boundary conditions on B. The difficulty lies in the fact that when Rm is no longer infinite then B is no longer zero within the body and B x and By at the surface are not known prior to the solution of the problem. The one thing we do know, however, is that when Rm  0 then B x and By should both  0 within the body. 30
37 Before resolving the above question of the boundary values of B x and By it will be necessary to introduce some order of magnitude arguments. It will be assumed that the thickness of the inviscid magnetic boundary layer, 6 i, is ) (Rm1/2 ), and that the thickness of the viscous sublayer, 6, is C9 (Re1/2 Also, differentiation with respect to x will not alter the order of magnitude of a quantity, while differentiation with respect to y will change the order of magnitude by 61. Whether it is 6 i or 6 v will depend on which layer is being discussed. It was noted earlier that B x and B tend to zero as Rm  , so that it will be valid to assume them both 0 (Rm ) at the wall. Since the viscous sublayer is extremely thin neither should vary appreciably from its value at the wall within this region. Next, consider the following relation xb=rm vxb (12.61) where v and B have been made dimensionless by reference values of v and B, where Ohm's law has been used, and E 0. Or, for the present two dimensional problem: 8B ab vb ax E8y x  Rm m2[ubyvbxj (12.62) Within the viscous layer the first two terms have the following order of magnitudes ab y (R 1/2) abx c (Pm 1/2 ax m n y ab so that aby/x << y. Also, within the viscous layer v << u and B and y ay y B x are of comparable order of magnitudes, so that vbx << U y. Accordingly equation (12.61) can be simplified, within the viscous layer to OBx  = R ub (12.63) Or, integrating over the viscous layer B x = B (x, o) + 0 (R/2 Re Bx = 0 (R 1/2) + 1/2 Accordingly, one can assume B x = 0 throughout the viscous sublayer and take this as the other inner boundary condition for the inviscid layer. 31
38 The equations governing both boundary layers will be deduced next, by similar order of magnitude arguments. Within the inviscid layer a~ a B R 1/25 ax + 8B = 0, so that assuming B =(1) leads to B = (R ax ay x y m Next, the two momentum equations (equation 8.17), and equation (12.62) are considered 0(i) 0 (i) 0(i) (i) C(i) auau 8P ( Bx IB x pu u + pv  IB By B '_+ (12.64) ( (R1 / 2) 0 (R 1/21 /2  9(R 1/2 m ~ m 0 ( m L C(m ) v +.+ pv B  y + B B y (12.65) p 8x ay ay x ax y ay S(Rm1/2) ((Rml/2) (R ml/ 0(R 1/2 8B  y R m uby  v Bx a~x ay m B 2 Bwo2 (12.66) where P= p +  and N = 'U. and it will be assumed that P and N are 0 (1). The resulting boundary layer equations in dimensional form for the inviscid layer are 8u 8u +8P =B x + pu a + p v 2+ ax Bx ax y ay (12.67) x7" a Ae (ub VBx (12.68) ax + a = 0 (12.69) ab x + ab Y = 0 (12.70) ax ay 32
39 with the boundary conditions at y = 0 u = u 00 (x) B x = B x0(x) auw(x) aty=0 v= 0 B =0 x It is interesting to see that a boundary condition on By at y = 0 is not needed since when By is eliminated from the equations by equation (12.70) they become second order in B x and there are two boundary conditions on B x available. value of B Y at y = 0 is part of the solution being sought and, accordingly, it is not a suitable boundary condition. From equation (12.65) it can also be seen that =(m ) and P = P (x) alone. Accordingly, L can be evaluated from the free stream solution. The boundary layer equations within the viscous layer are precisely those obtained earlier in the present chapter. The applied magnetic field, By, is taken to be the value found at y = 0 from the inviscid boundary layer solution. The pressure, p, to be used is P(x) obtained from the potential flow solution. In conclusion then, when Rm  cc to conventional boundary layers can exist. of new features for future studies. (8) boundary layer phenomena very similar They do, however, offer a number The 33
40 REFERENCES 1. Ludford, G. S.S. Rayliegh's Problem in Hydromagnetics: The Impulsive Motion of a Pole Piece. Archiv for Ratioral Mechanics and Analysis, 3, 14 (1959). 2. Rossow, V.J. On Flow of Electrically Conducting Fluids Over a Flat Plate in the Presence of a Transverse Magnetic Field. NASA Report 1358, Sherman, A.. Viscous Magnetohydrodynamic Boundary Layer. Phys. Fluids 4, 552 (1961) 4. Howarth, L. Ed. Modern Developments in Fluid Dynamics. High Speed Flow, vol. 1 Oxford University Press, Bush, W. B. Compressible Flat Plate Boundary Layer Flow with an Applied Magnetic Field. J. Aerospace Sci. 27, 49 (1960) 6. Kerrebrock, J. L. Electrode Boundary Layers in Direct Current Plasma Accelerators. J. Aerospace Sci. 28, 631 (1961). 7. Bush, W. B. The StagnationPoint Boundary Layer in the Presence of an Applied Magnetic Field. J. Aerospace Sci. 28, 610 (1961) 8. Sears, W. R. On a Boundary Layer Phenomenon in MagnetoFluid Dynamics. Astronautica Acta 7, 223 (1961). 34
41 SEN ERAL 0 ELECTRIC SPACE SCIENCES LAUORATORY MISSILE AND SPACE DIVISION TECHNICAL INFORMATION SERIES AUTN SUBJICT CLASSIFICATIO NO. Rb3SD4 A. Sherman Magnetohydrodynamics DATE Jan TinL CLASS MAGNETOHYDRODYNAMIC I BOUNDARY LAYER CLANNoV. None WWuWW C, V PM l AT OM umw NO. 1 M1PS ww.? VAUSY PoaE PACI 37 WWf COW~ RMO PlIUANA PA. A review of those new phenomena which arise in magnetohydrodynamic boundary layers is presented in this report. The first topic discussed due to its simplicity and relation to boundary layer flows is the Rayleigh problem. Here it is shown that the introduction of magnetohydrodynamic forces leads to basic changes in the nature of the flow. Next the basic boundary layer equations are deduced and the conditions necessary to yield similar solutions deduced. In succeeding sections incompressible and compressible boundary layers are treated with attention paid to the appropriate external boundary conditions. Finally, boundary layer phenomena arising when Rm C are considered in order to illustrate some new phenomena within the framework of boundary layer theory. It is intended that the present material will be one chapter of a forthcoming book. sy sog u *is ole omd Mldig on Obo *ems rn.. wbs ebove lademogto ame 10 fud We a o0andsed aid Olf. AUINI( D fil